Theory of the Collapsing Axisymmetric Cavity

We investigate the collapse of an axisymmetric cavity or bubble inside a fluid of small viscosity, like water. Any effects of the gas inside the cavity as well as of the fluid viscosity are neglected. Using a slender-body description, we compute the local scaling exponent α=dln⁡h₀/dln⁡t^′ of the minimum radius h₀ of the cavity, where t^′ is the time from collapse. The exponent α very slowly approaches a universal value according to α=1/2+1/[4√-ln⁡(t^′)]. Thus, as observed in a number of recent experiments, the scaling can easily be interpreted as evidence of a single nontrivial scaling exponent. Our predictions are confirmed by numerical simulations.